What causes this very dark areas, that can't be shadows?
This can be the result of overlapping geometry - causing Z-fighting where two or more faces are in the exact same location and orientation. This can result in the faces shadowing each other and other artefacts  such as banding.
Check your meshes and objects to see if you have a duplicate mesh at the exact same point (try selecting one and move or hide it to see if the other remains) or whether your mesh has multiple overlapping faces (move a vertex and see if another face is in the same place).